1. 5.4 Sampling Distributions and the Central Limit Theorem Key Concepts: –How to find sampling distributions and verify their properties –The Central Limit Theorem –Using the Central Limit Theorem to find probabilities

2. 5.4 Sampling Distributions and the Central Limit Theorem What is a sampling distribution? –The probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. Examples of sample statistics:

3. 5.4 Sampling Distributions and the Central Limit Theorem Properties of sampling distributions of sample means: 1. The mean of the sample means,, is equal to the population mean : 2. The standard deviation of the sample means,, is proportional to σ : (valid when n ≤ 0.05N) Note: is known as the standard error of the mean.

4. According to Forbes Magazine, the top five wealthiest women in the world in 2010 were: Christy Walton: $22.5 billion Alice Walton: $20.6 billion Liliane Bettencourt: $20.0 billion Birgit Rausing: $13.0 billion Savitri Jindal: $12.2 billion 1. Let X = wealth (in billions of dollars). Find the mean and standard deviation of X.

5. 5.4 Sampling Distributions and the Central Limit Theorem 2. List all possible samples of size 2 from this population of 5 and list the sample mean for each sample. 3. Find the mean and standard deviation of the sample means in column three. SampleWealth Sample Mean {C, A }22.5, 20.621.55 { C,L }22.5, 20.021.25 { C,B }22.5, 13.017.75 { C,S }22.5,12.217.35 { A,L }20.6, 20.020.3 { A,B}20.6, 13.016.8 { A,S }20.6, 12.216.4 { L,B }20.0, 13.016.5 { L,S }20.0, 12.216.1 { B,S }13.0, 12.212.6

6. 5.4 Sampling Distributions and the Central Limit Theorem Where does the Central Limit Theorem fit in? –We use the CLT to make a statement about the shape of the distribution of the sample means (see p. 263) If samples of size 30 or more are drawn from any population with mean µ and standard deviation σ, then the sampling distribution of the sample means will be approximately normal. If the population itself is normally distributed, then the sampling distribution of the sample means is normally distributed for any sample size n.

7. 5.4 Sampling Distributions and the Central Limit Theorem Practice using the Central Limit Theorem #10 p. 270 (Annual Snowfall) #24 p. 271 (Canned Vegetables) #30 p. 272 (Gas Prices: California) #36 p. 272 (Make a Decision)